on the structure of bernstein algebrasbrusso/cortesmontanerjlms95bis.pdfsections 2 and 3 suggest the...

ON THE STRUCTURE OF BERNSTEIN ALGEBRAS

TERESA CORTES AND FERNANDO MONTANER

0. Introduction

Bernstein algebras were introduced by Holgate [6] as an algebraic formulation ofthe problem of classifying the stationary evolution operators in genetics (see [8]).Since then, many authors have contributed to the study of these algebras, and thereis a fairly extensive bibliography on the subject. Known results include classificationtheorems for sorre types of Bernstein algebras (for instance, Bernstein algebras ofdimension less than or equal to four, see [12, 9, 2]) as well as some other structuralresults (see for instance [1, 5, 11, 4]). However, due to the intrinsic complexity ofBernstein algebras, there is not a general structure theory for these objects which mayserve as a starting point for a general classification theorem.

In this paper we contribute to such a theory through two main ideas, directproducts and Bernstein-Jordan algebras. We define direct products of Bernsteinalgebras in Section 2, together with the related notions of decomposable andindecomposable algebras, and mention some of their properties, including aKrull-Schmidt Theorem of uniqueness of such decompositions. In Section 3 we dealwith Bernstein-Jordan algebras. This subject has been considered by several authors,and we collect here some known results. We also show that being Jordan is not justan artificial condition for a Bernstein algebra. A class of Bernstein-Jordan algebraswhich we call reduced, arise naturally as homomorphic images of Bernstein algebras,and in fact, a functor from the category of Bernstein algebras into the category ofBernstein-Jordan algebras can be defined.

Sections 2 and 3 suggest the following approach to the structure of Bernsteinalgebras: the first step is to take a quotient algebra which is reduced, then todecompose the resulting algebra as a direct sum of indecomposable algebras andfinally to classify the indecomposable Bernstein-Jordan algebras obtained in thatway. In Section 4 we show that an obstacle to this approach appears at the last step.We give examples of infinite families of indecomposable Bernstein-Jordan algebrasthus showing that the classification of these algebras presents some serious difficulties.

1. Preliminaries

Throughout this paper O will denote an infinite field of characteristic not two.A finite dimensional commutative algebra A over the field O together with a

homomorphism of algebras (weight homomorphism) CQ:A^Q> is called a Bernsteinalgebra if every xeA satisfies

(x2)2 = co(xfx\

Received 18 September 1992.

1991 Mathematics Subject Classification 17D92.

Research partially supported by the DGICYT (PS 90-0129).

J. London Math. Soc. (2) 51 (1995) 41-52

42 TERESA CORTES AND FERNANDO MONTANER

Next we recall some known results that can be found in [12].For any Bernstein algebra A, the weight homomorphism is uniquely determined.Each Bernstein algebra possesses at least one idempotent, and to each idempotent

e e A is associated a Peirce decomposition A = <t>e + Ue + Ve, where

Ue = {xeA\2ex = x} and Ve = {xeA \ex = 0},

and Kerco = Ue+ Ve. The Peirce subspaces multiply according to

ueve^ue, v2e^ue, ui^ve, uev

2e = o.

The set of idempotent elements of A is given by \{A) = {e + u + u2\ueUe} where eis any given idempotent of A. For two idempotents e a n d / = e + u + u2, we have thefollowing relations between the corresponding Peirce spaces Uf = {x + 2xu\xeUe}and Vf = {x — 2x(u + u2)\xe Ve}. This shows that the numbers dim Ue and dim Ve donot depend on the idempotent e. The pair (dim Ue+\, dim Ve) is called the type of A.

2. Direct products of Bernstein algebras

To define the direct product of a family of Bernstein algebras the natural settingis the category of baric algebras over a field O, whose objects are the pairs (A,co)where A is a (not necessarily associative) algebra and co:A^-<t> is a nonzerohomomorphism (of algebras) called a weight homomorphism. In this category,morphisms between {Ax, coj and (A2, co2) are defined as homomorphisms of algebras<fi:Al->A2 satisfying the condition co1 = co2<j>, that is, making commutative thediagram

These will be called morphisms of baric algebras or baric morphisms.Homomorphic images are associated to (baric) ideals, ideals / of A contained in

Kerco. For these ideals the factor algebra A/1 is baric with the inducedhomomorphism co':A/I-+Q>, given by co'(x + I) = co(x). Throughout this paper, idealwill mean baric ideal. In the same way, subalgebras of baric algebras will beunderstood to be subalgebras not contained in the kernel of the weighthomomorphism (so that they are also baric algebras with the restriction of theoriginal weight). It is straightforward that with these conventions any homomorphismof baric algebras <f>:(A1,co1)-+(A2,co2) induces an isomorphism of baric algebrasAJKerfi ^ Im0.

Inside the category of baric algebras we consider the category & of Bernsteinalgebras, that is commutative baric algebras (A, co) satisfying Bernstein identity

(JC2)2 = co(x)2x2.

Notice that the uniqueness of the weight homomorphism implies that anysurjective homomorphism <f>:(A1,co1)^'(A2,co2), of two Bernstein algebras is, in fact,a morphism of baric algebras.

The simplest example of a Bernstein algebra is the ground field O with the identityhomomorphism. This is a final object in @. Also, the existence of idempotent elements

ON THE STRUCTURE OF BERNSTEIN ALGEBRAS 43

means that any Bernstein algebra has a subalgebra isomorphic to <D (although it maynot be unique). We shall say that a Bernstein algebra is trivial if it is isomorphic to<t>.

Let (y44,c0()(e/ be a family of Bernstein algebras. It is easily seen that the directproduct n<e/^< is n o t a Bernstein algebra i f /has at least two elements. We considerin ]~[(6/-^t the set

x i e / (At, cot) = {(xt)e Yliei At | (£>lxt) = cofa) for all ije/}.

Since all the maps co{ are homomorphisms of algebras, it is clear that X ieI(At, co^)is a subalgebra of flte/^i- We can define a weight homomorphism on Xi6/(^(,co1)be setting co((xj) = cot(xj which is clearly independent of the index i. Then it isstraightforward that (xi e l (Ai ,coi) , co) is a Bernstein algebra. The canonicalprojections of n<e/^< o n t o t n e factors induce projections n^. xieI(A0 co^{Apco^),and since coj n^(xt)) = cofa) = co((x()), the map ni is a baric morphism.

PROPOSITION 2.1. In the category 38, the algebra { XieI(Ai,coi), TTJ is the directproduct of the family (A0 cot)ieI.

Proof. Let <j>i: (C, T) -+ (At, cot) be morphisms of baric algebras. We define^(c) = ( ^ ( ( c ) ) e f ] ( 6 ; ^ Thus <j>:C->Y\ieiAi is a homomorphism of algebras.Moreover co( 0( = T, hence for all indexes /, j we have cot <pt = co} <j)p and this impliesthat 0(C) is contained in X ie[ (At, cot). Thus we can consider 0 as a homomorphism0:C->- XieI(At,cot). Clearly TT(0 = <pt for all /, and axfi = co(7rt0 = coi^>l. Therefore,since cot <f>t = T, this shows that 0 is baric. Finally the uniqueness of 0 is clear.

EXAMPLE 2.2. Let A = d>e + d>u + d>v be the commutative algebra with e2 = e,leu = u, u2 = v, and all other products equal to zero. Then A is a Bernstein algebrawith weight homomorphism co(oie+/3u + yv) = a, and Ax A is isomorphic to thecommutative algebra (be + (^^ + Ot^ + Q>u2 + Q>v2 with u\ = vx, u\ = v2, e2 = e,2eux = ult 2eu2 = u2, and all other products equal to zero.

Having a notion of direct product we can introduce the related notions ofdecomposable and indecomposable algebras. Notice first that the trivial algebra <D isalways a direct factor of any algebra since clearly <P x A is isomorphic to A. Thus, todefine decomposable algebras we must ask for nontrivial factors. We say that aBernstein algebra (A,co) is decomposable if there exist algebras (At, coj and (A2, co2),both nontrivial, such that (A,co) = (Al,co1)x(A2,co2). We shall say that (A,co) isindecomposable if it is not decomposable.

Decomposability may be characterized in terms of (baric) ideals as follows.

LEMMA 2.3. A Bernstein algebra (A, co) is decomposable if and only if there arenonzero ideals Ix, I2 of (A, co) such that Kerco = / 1 © / 2 as algebras. In this case

Proof. Let 0 be the isomorphism (A,co) = (A1,col)x(A2,co2) and nt thecorresponding projection for / = 1,2. Then clearly Kerflx0 = 1X and Ker;r20 = I2

satisfy Kerco = /2 © I2 and they are nonzero ideals of (A,co).Conversely, the canonical maps/?,: A -» A/^ are baric (since the ideals Ix and I2 are

baric), hence they induce a morphism (f>\A->A/Ix x A/I2. Since IX(\I2 = 0, it followsthat 0 is injective, and Kertw = Ix + l2 implies that 0 is surjective.


Direct factors of an algebra A can be considered as subalgebras of A.

LEMMA 2.4. A Bernstein algebra (A, co) is decomposable if and only if there existnontrivial subalgebras A{, for i = 1,2, of A such that (Al ft Ker co)(A2 0 Kera>) = 0,Ax + A2 = A, and A1 0 A2 is a trivial algebra. In this case A = Axx A2.

Proof. Let <f> be the isomorphism (A,co) = (B1,col)x(B2,co2) and ft, thecorresponding projection for / = 1,2. As in Lemma 2.3, Ker co = Ker nx <j> 0 Ker TC2 <f>.Now if e is any idempotent of A, then Ax = <be + Ker n2 <f> and A2 = <&e + Ker nx <j> aresubalgebras of A which satisfy the required conditions.

Conversely, if Ax, A2 satisfy the above conditions, setting It = Ai n Kereo givesideals 7X and I2 satisfying the conditions of Lemma 2.3, and A/Ix = A2, A/I2 s Av

Let A be a finite dimensional Bernstein algebra. If A is not indecomposable, thenA = A1xA2isa. product of Bernstein algebras. If any of the A{ is not indecomposable,we can further decompose it. The finite dimensionality of A implies that thisprocedure reaches an end. Thus we have the following.

PROPOSITION 2.5. Every finite-dimensional Bernstein algebra is a direct product ofa finite number of indecomposable Bernstein algebras.

It is natural to ask about the uniqueness of this decomposition. Note first that ingeneral we will not have uniqueness of the factors as subalgebras of A. This is easilyseen in the following example. Let A = <&e + Owx + Q>u2 with 2eu{ = ut and all otherproducts zero. Then

A = (Oe + OMJ x (Oe + <Du2) = (Oe + <t>{ux + u2)) x (Oe 4- <D(«X - u2)).

Therefore uniqueness should take place in a weaker sense. This is what the followingKrull-Schmidt Theorem for Bernstein algebras shows.

THEOREM 2.6. Let (A,co) be a finite dimensional Bernstein algebra. Assume thatA is not the trivial algebra and let

(A, co) £ (Av 0)0 x ... x (An, con) £ (Bv T J X ... x (Bm, T J

be two decompositions of (A, co) as a direct product of indecomposable nontrivialBernstein algebras. Then n = m, and after a reordering of the indexes, (Ao cot) =

This result was first presented by the second author in a talk given in theInternational Conference on Nonassociative Algebraic Systems held at the Universityof Malaga in 1991. Recently, it has also been proved (independently and in the moregeneral setting of baric algebras) by R. Costa and H. Guzzo [3]. Here we shall giveonly the main ideas involved in the proof of this theorem and refer to [3] for acomplete proof (which is essentially the same as the one given originally by the secondauthor). We shall use these ideas later to prove a stronger version of Theorem 2.6 underadditional conditions on the algebra (A, co).

Theorem 2.6 is based on the corresponding result for modules over a ring. Touse these ideas we recall the following definitions.

If A is any (nonassociative) algebra, we denote by M(A) the multiplication algebraof A, that is the subalgebra of E n d ^ ) , the ring of endomorphisms of the vector


space A, generated by the identity map and all operators Rx, Lx of left and rightmultiplication by elements xeA. Then A can be considered as an M(^)-module, andan M(/4)-submodule of A is the same as an ideal of A. In particular, if (A,co) is aBernstein algebra, the ideal Ker co is a submodule of A. Lemma 2.3 shows that thedecomposability of A is equivalent to the decomposability of the module Ker co.

LEMMA 2.7. Let {A, co) be a Bernstein algebra, then A is decomposable if and onlyi /Kera) is a decomposable M(A)-module.

If (A, co) is a homomorphic image of (B, T) we can relate the module structures ofKer co and Ker t by means of the following.

LEMMA 2.8. Let y/:(B,T)-^(A,co) be a surjective morphism of two Bernsteinalgebras. Then y/ induces an epimorphism \p: M(B) -> M(A), thus giving to Ker co anM(B)-module structure. With this structure the restriction ofy/, y/0: Ker x -> Ker co, is anepimorphism of M(B)-modules.

Proof. Any element from M(B) is of the form f[Rx,...,RXk) where / is apolynomial in k noncommuting variables and xx,...,xk are elements of B. Letj{Rx,..., Rx) = g(RVi,..., Ry) be two representations of an element of M(B). If x e B,we have/(/?Zi,..., RXk) x = g(Ry ,..., Ryi) x. These are products of elements of B, henceapplying ^ , 'we have f{R¥iXi),\..,R¥{Xk))y/(x) = g{Rw(Vi), ...,R¥lyJy/(x). Since yr issurjective, this implies that KR^, . . . ,-K^)) = giRv(Vl), . . . , ^ ( t f | ) ) - Therefore thecorrespondence (p(j{Rx{) ...,RXk)) =f{Rv(x^, ...,Rv(Xk)) defines a homomorphism. Theabove equality also shows that the restriction y/Q: Kerr->Kerco is an epimorphism ofM(i?)-modules.

REMARK 2.9. Let (A{, cot)ieI be a family of Bernstein algebras, and let (A, co) beits direct product. By Lemma 2.8 Ker<w( is an M(.4)-module via the canonicalprojection nt: A -> At, and it induces a homomorphism of M(y4)-modules Ker co ->•

It is clear that these maps give rise to an isomorphism of M(^4)-modules^fkg jKerav Notice that Lemmas 2.7 and 2.8 imply that if (A,co) and (B,r)

are Bernstein algebras and (A, co) is an indecomposable homomorphic image of(B, r), then Kerco is an indecomposable M(i?)-module (with the induced structure).

The proof of Theorem 2.6 is based on the fact that in that situation we have aninduced M(y4)-isomorphism between ]^[ie/Kerco( and P[ j 6 JKerT; and we can applythe Krull-Schmidt Theorem for M(y4)-modules. In fact we can consider the modulesKer co, and K e n , as submodules (hence ideals) of Kerco. Under this identificationKer co = ® < e / Kerco( = ®jeJ Ker r r We have the following consequence ([7], cf. also[3]) of the Krull-Schmidt Theorem which will be used later.

COROLLARY 2.10. Under the conditions of Theorem 2.6, after a suitable reorderingof the indexes, we have for all k = 1,...,«, the following decompositions of the M(A)-module Ker a;

Ker co = K e r c ^ ® ... 0Kercofc©KerTfc+1© ... © Kerrn.

To close this section we give some examples of indecomposable Bernsteinalgebras.


EXAMPLES 2.11. Any Bernstein algebra of dimension two is indecomposable. Asfor dimension three, from the classification of these algebras (see [12]) we obtain thefollowing examples of indecomposable algebras. Here A = Oe + <Dw + <3>v and wewrite only the nonzero products:

(1) e2 = e, leu = u, u2 = v,(2) e2 = e, leu = u, uv = u,(3) e2 = e, leu = u, v2 = u.

Four-dimensional Bernstein algebras over an arbitrary infinite field of charac-teristic not two have been classified by the first author [2]. We refer to the algebrasas they are numbered in that paper. There, the indecomposable algebras are (3), (5),(7) and (8) among those of type (2, 2), and (10), (12), (15), (16), (17), (18), (19), (20)and (21) among those of type (3, 1).

Over any infinite field there exists an infinite family of indecomposable algebrasof dimension 4. Namely, the family (19) in the classification of [2]: algebrasA = Q>e + Oi/j + <Dw2 + Q>v with nonzero products leut = w<? uyv = ux, u2v = piu2 for//ed>*. As shown in [2], two of these algebras with parameters ji, /i' are isomorphicif and only if either // = //' or //// = 1.

Over an algebraically closed field that is the only infinite family of indecomposableBernstein algebras of dimension 4. These are not Jordan algebras, hence in particularthere are finitely many indecomposable Bernstein-Jordan algebras of dimension 4.

The importance of Bernstein-Jordan algebras in the study of Bernstein algebrascomes from the fact that, as we shall show in the next section, they arise naturally asfactor algebras of general Bernstein algebras. In this sense Bernstein-Jordan algebrasplay the role of semisimple algebras for a radical class. It would be desirable that thesituation encountered with algebras of low dimension, that is the existence of a finitenumber of indecomposable algebras over an algebraically closed field, extended to allBernstein-Jordan algebras. However, we shall see in the last section that, alreadyin dimension 7, there are infinitely many non-isomorphic indecomposableBernstein-Jordan algebras.

3. Bernstein-Jordan algebras

In this section we show how Bernstein-Jordan algebras can be obtained naturallyfrom Bernstein algebras. This reduction has the advantage that Bernstein-Jordanalgebras can be more easily studied. However, the structure theory of Jordan algebrasdoes not apply directly to the Bernstein case since, apart from the results dealing withnilpotency and related concepts, that theory is mainly focused on the description ofalgebras which satisfy some regularity condition such as primeness or simplicity, andthese conditions imply the triviality for a Bernstein algebra (see [5]).

We start with several characterizations of Bernstein-Jordan algebras which havebeen found by Worz-Busekros [13], Walcher [11], Ouattara [10], Alcalde, Baeza andBurgueno [1], Gonzalez and Martinez [4].


PROPOSITION 3.1. For a Bernstein algebra {A, co) the following assertions areequivalent:

(1) A is a Jordan algebra;(2) V2

e = 0foralleeI(A);(3) V] = 0 and for all ueUe,veVe we have (uv)v = Ofor all eeI(A);(4) V2 = 0 and for all ueUe, veVe we have (uv)v = Ofor some eeI(A);(5) A satisfies the identity x3 = co(x) x2.

Let A be a Bernstein algebra. If /, J are ideals of A such that A/1 and A/J areJordan algebras, then A/If\J is also a Jordan algebra. We denote by J(A) the leastideal of A such that A/J(A) is a Jordan algebra. Since A/Kerco ^ <I> is a Jordanalgebra, it is clear that J(A) is a baric ideal. The above characterizations ofBernstein-Jordan algebras imply the following characterizations of J(A) (here idenotes the ideal of A generated by X):

eeHA)

Next we prove some basic properties of J(A).

LEMMA 3.2. Let A be a Bernstein algebra, S be a subalgebra of A and I be an idealof A. Then

(l)

Proof (1) follows immediately from the equality

J(A) = idA({x3-co(x)x2\xeA}).

To see (2) set J(A/I) = H/I for an ideal H of A. Then (A/I)/(H/I) s A/H is aJordan algebra, hence J(A) £ H and J(A) + I/I £ J(A/I). Now

is a Jordan algebra, hence J(A/I) £ J(A) + I/I.

Notice that in Lemma 3.2 (1) we do not have the equality in general. Consider forinstance the commutative algebra A = <be + <£« + Ot> with 2e« = M and v2 = «, and allother products equal to zero, and the subalgebra S = <be + <X>w, which is, in fact, anideal of A. Here J(5) = 0, J(A) = ®u^S.

We show next that the correspondence A^-A/J(A) is functorial.

THEOREM 3.3. Let A, B be Bernstein algebras and let <f>:A^-B be a baricmorphism. Then (j>{i{A)) c= J(2J) and the induced map <f>:A/J(A)^-B/J(B) is a baricmorphism. Thus the correspondence F:A^A/J(A) defines a functor from $ into theclass of Bernstein-Jordan algebras. The functor F preserves finite direct products.

Proof. Let 0':,4/Ker0-*Im0 be the isomorphism induced by 0. Then,by Lemma 3.2 we have <f>(J(A)) = <j>'(J(/4) + Ker0/Ker 0) = f(J(,4/Ker0)) =J(Im 0) c ](B). Thus <f> induces a homomorphism <j>: A/J(A) -> B/J(B). Since J( 4) andJ(5) are baric ideals, this is a morphism of baric algebras. Clearly the composition of


maps is preserved by passing to the factor algebras, hence F:A^*A/J(A), F(<f) = <j>defines a functor.

Now, if (A^cOi) for iel is a finite family of Bernstein algebras, let (A,co) =X^jiA^co^. Now by the above characterizations of J we have J(A,co) =idA({x3 — co(x)xz\x6A}) = idA((x*-coi(xi)Xi)\(xi)eA}. Since the product in (A,co)is componentwise and *?—a^x,)*? belongs to Kerco, for all i, it follows thatJ(A,co)^YlieIidAi({x*-coi(xi)x*\xi€Ai}) = Y\ieIJ(Ai). A s / i s finite, the reverseinclusion is also easily proved. Thus

The ideal J(A) may be difficult to handle, however it can be bounded by an idealeasily obtained from any Peirce decomposition of A. In fact, if A = <Pe+Ue+Ve isthe Peirce decomposition of A associated to the idempotent e, the set U0(A) ={ue Ue\uUe = 0} is an ideal of A, and A/U0(A) is a Jordan algebra. This has beenproved by Hentzel and Peresi [5] and by Gonzalez and Martinez [4]. It is proven alsoin [4] that U0(A) does not depend on the choice of the idempotent e. We obtain thisfrom another characterization of U0(A).

PROPOSITION 3.4. Let A be a Bernstein algebra. Then f\eel(A) Ue = U0(A), and itis an ideal of A such that A/U0(A) is a Bernstein-Jordan algebra.

Proof See [5, 4]. The only assertion which remains to be proved is the equalityU0(A) - n«eiu) Ur Let e, fel(A), then / = e + x + x2 for some x€Ue. The Peircespaces Ue and Uf satisfy Uf = {u + 2xu\ UE Ue}. Now if ue Ue 0 Uf, then u = u' + 2xu'for some u'eUt. Thus u-u' = 2xu'eUe0 U] £ uen Ve = 0. Since I(^) ={e + x + x*\xeUe}, then uef)eel(A)Ue implies that ux — 0 for all xeUe, hence

ueU0(A)-On the other hand, if we U0(A), then u = u + 2uxe Ue+X+Xi for all xe Ue. Hence

COROLLARY 3.5. Let A be a Bernstein algebra, then J(A) c f|eeI(i4) U, andJ(A)2 = 0.

Next we prove some other properties of U0(A). We shall make use of the followingstraightforward result.

LEMMA 3.6. Let A be a Bernstein algebra and let I be an ideal of A. Then for anyidempotent e of A with associated Peirce decomposition A = <&e+Ue+Ve, we haveI = I()Ue + I()VeandA/I^®e+ UJ(I n Ut) + VJ{I n Ve) with products

= \u=0,= u*

(v+(inve)f =y2

ue))(v+(in ve)) = uv+(in ue).


PROPOSITION 3.7. Let A be a Bernstein algebra and let I be an ideal of A. Ifn: A^-A/I is the canonical projection, the following assertions are equivalent:

(1) /c£/0(4(2) for all xeA, we have that n(x) is idempotent if and only if x is idempotent.

Proof. Assume first that / £ U0(A). Let e be any idempotent of A and letA = <!>e + Ue + Ve be its associated Peirce decomposition. If n(x) = x + Iis idempotent,set x = e + u + ve$>e+Ue+Ve. Now x2-x = u2 + 2uv-v + v2eU0(A) c Ue. Thus,matching components in the Peirce decomposition, v = u2, and x = e + u + u2 is anidempotent.

Conversely, let e€\{A). If veld Ve, then by Lemma 3.6, n(e + v) = n(e) is anidempotent in A/1. Hence by (2) e + v is an idempotent in A. But this implies thatv = 0. Thus If] Ve = 0, and / £ Ue. Since the idempotent e was arbitrary, / ^ U0(A).

We shall say that a Bernstein algebra A is reduced if U0(A) = 0. We next show thatfactoring out the ideal U0(A) gives a reduced algebra.

PROPOSITION 3.8. If A is a Bernstein algebra, then A/U0(A) is reduced.

Proof. Take/an idempotent of A/U0(A) and let e+ UQ(A) =/with eeA. ByProposition 3.7 e is an idempotent and we have the Peirce decompositionsA = <De+ Ue+ Ve and A/I = 0>/+ £/,+ Kr Now by Lemma 3.6, Uf = UJU0{A), hence

U0(A/U0(A)) = D/eiM/u^)) tf, = (n.eiW> t/J/tfoW = U0(A)/U9(A) = 0.

4. Direct products of reduced Bernstein algebras

In this section we prove a stronger version of Theorem 2.6 for reduced nuclearBernstein algebras. We show that the uniqueness asserted in Theorem 2.6 can bestrengthened to the uniqueness of the kernels of the indecomposable subalgebras thatare direct factors of the decomposition. After that result we give some examples ofinfinite families of indecomposable reduced Bernstein algebras.

Recall that a Bernstein algebra A is called nuclear if A2 = A. In terms of the Peircedecomposition A = Oe+ Ue+ Ve associated to an idempotent e of A, this means thatUl = V,

Let (A, co) be a Bernstein algebra and assume that we have a decomposition(A, coj^iA^coJx ...x(An,con). Through this isomorphism the kernels Kercot

correspond to ideals /, of A and Kerco = Ix® . . .©/„ (see Remark 2.9). We call thisthe associated decomposition of Ker co.

THEOREM 4.1. Let (A,co) be a finite dimensional reduced nuclear Bernsteinalgebra, and let (A,co) £ (A^co^x ... x(An,con) s ( ^ T ^ X ... x(Bm1xm) be twodecompositions of(A,a>) as direct products of indecomposable Bernstein algebras, withassociated decompositions o/Kerco:

Then « = m, and after a reordering of the indexes, It = Jv


Proof. By Corollary 2.10, we have n = m and, after a reordering of the indexes,we have decompositions of Kerco:

Kerw = / 1 © . . . 0 / f c © / J k + 1 0 . . . © y n for k=l,...,n.

Note first that this implies that /, Ji — 0 for any j > i. Thus, for any /, we haveJ\ = J{ Ker co = Jt{Ix 0 ... 0 /, 0 Ji+10 ... 0 Jn) = Jt Iv Therefore

/, ./,<=/, fl {J, Ker co) s /( n /, = 0,

if / #y. Finally, / 2 = /,Kerco = / ^ 0 ... 0 /„) = IiJi = /» , for any i.Now, if Are/,, then x = >>! + ...+.)>„ with ^ e / r Thus for any z e / , , we have

{x-y^)zeIiJjJtJiJj = 0 if / #y, and if z e / , , then

Hence /, ^ 7, + Ann (Ker co), where Ann (Ker co) = {x e Ker co\ x Ker co = 0}.Now let <? be any idempotent of A. By Lemma 3.6 we have /, = /, n Ue + It 0 Ve for

all / = 1, ...,n. Since A is reduced, Ann (Ker co) C\ Ue = 0, hence multiplying by e, theinclusion /< £ J{ + Ann (Ker co) gives /4 n Ue £ / t fl C/e. Since ,4 is nuclear, Ke = V\.Hence

Therefore /, = 7, n £/e + /( fl Ke s; / t . Since they have the same (finite) dimension, theyare equal.

Obtaining reduced nuclear Bernstein algebras is easy. First, to get a nuclearBernstein algebra from a Bernstein algebra A it is enough to consider the square A2.On the other hand, Proposition 3.8 shows that A/U0(A) is always reduced. Moreover,these two operations commute: (A/U0(A))2 = A2/UQ(A2). Hence the resulting algebrais both reduced and nuclear.

EXAMPLES 4.2. Next we give two examples of infinite families of indecomposablereduced Bernstein algebras. The existence of such examples leaves little hope forcompleting the structure theory exposed in the preceding sections.

1. Set Ak = Oe 4- ("x, ...,«„) + Ou, the commutative algebra with e an idempotentand Ue = O(M15 ...,«„), Ve = <J>v the corresponding Peirce spaces. We set v2 = 0,utv = 0, u2 = v for all / and utu} = 0 if / ^ j . If / i s an ideal of Ak with If\Ue^0, thenit contains an element of the form Xx ux + . . . + Xn un with some Xt ^ 0. ThenX(v = (Xiu1 + ...+Xnun)uteI, hence vel. Thus any nonzero ideal of Ak contains theelement v so that Ak is indecomposable by Lemma 2.3. It is clear that it is alsoreduced.

2. On the vector space with basis e, ux, u2, u3, u4, vx, v2 over an algebraically closedfield d>, we define the commutative algebra A(<x,jl) for a, jSeO, with multiplicationtable e2 = e, eut = |«<5 evi = 0, u\ = vx, u\ = v1 + vi, ul = vl + <xv2, ul = v1+pv2,u^ = 0 for / j£j, vtv} = 0 and uivi = 0.

(a) If a # fi and a, $ # 0, 1, then y4(a, /?) is indecomposable and reduced. Indeed,let /, J be nonzero ideals of A(a, ft) with I+J = Kerco. Since dim Ue = 4, we canassume that dim(/n Ue) ^ 2. Take a nonzero element Xlu1 + X2u2 + X3u3 + /I4«4e/.


Multiplying this by ux, u2, u3 and w4, we get that Xxvx, X2{vx + v2), X3(v1 + av2) andX4(v1+^v2) belong to /. Since dim(/n Ue) ^ 2, we obtain that vx + yv2, v1 + dv2el forsome y # 3. Thus Ve = <X>(ul5 v2) s /. This clearly implies that IftJÔ. Hence byLemma 2.3 A{cn,p) is indecomposable. It is straightforward that A(<x,fi) is reduced.

(b) Let <fi:A(oi.,fI)Â((x',P') be an isomorphism between two of these algebras.Notice that for any idempotent/ey4(a,y9), we have Ve= Vf and Uf has a basis zl5 z2,z3, z4 such that z\ = u15 z2 = vx + v2, z\ = vl + txv2, z\ = vx+^v2, zizi = 0 for / #y,u(i^ = 0 and ztt^ = 0 (take zt = ui + 2uiu, where f=e + u + u2, and note thatztz^ = UfUj). We shall call this a standard basis for A(<x,fi).

Now 0(e) = / i s an idempotent of A(a', f}') and 0(f/e) = C//5 0(ê) = Vf. Fix e, M15

«2> M3> M4> yi» y2 a standard basis of A{OL, P), and/, z15 z2, z3, z4, wx, w2 a standard basis

of A(<x',fi'). Let A = (Ay) be the matrix of the restriction <f>:Ue->Uf and let</>(v1) = y1w1 + y2w2, (/>(v2) = S1w1 + S2w2. Setting (al5 a2, a3, a4) = (0, l ,a , /0 and0^l5 A. s» A) = (0 . ! '« ' ' ^') w e n a v e w? = v1 + <xiv2, so that

^("(2) = (7i + «i <5i) wx + (y2 + oct S2) w2,

M) = 0W = (E W =

And if / # j we have

0 = 0(w{ M,) = ^ £

Therefore, setting5 = Diagonal 0?!, #,, ^3, y94),

Df = Diagonal (yt + ax (5i5 yi + a2 <5(, yt + a3 3t, yt + a4 St) for / = 1, 2,

we have A/V = Dx and A-fiA* = D2. On the other hand A is invertible, hence so is Dv

Thus A( = A~1D1 and A5A"1/)! = D2. Hence

The eigenvalues of 5 must equal the eigenvalues of the matrix on the right-handside of this formula. Hence

Since the transformation x-+(y2 + xd2)/(y1 + xdl) preserves the cross ratio a offour numbers, the set of values obtained by taking the cross ratio of {1,0, a, /?} in anyorder gives only a finite number of algebras possibly isomorphic to the originalalgebra A{<x,f$). Since the cross ratio of (1,0, a, /?) can take any value of <X> — {1,0} whena # P and a, ft # 0, 1, and <D is infinite, it is clear that there exist infinitely manynonisomorphic algebras of the form A{cn,p).

References

1. M. T. ALCALDE, R. BAEZA and C. BURGUENO, 'Autour des algebres de Bernstein', Arch. Math. 53(1989) 134-140.

2. T. CORTES, 'Classification of 4-dimensional Bernstein algebras', Comm. Algebra 19 (1991) 1429-1443.

52 ON THE STRUCTURE OF BERNSTEIN ALGEBRAS

3. R. COSTA and H. Guzzo JR., 'Indecomposable baric algebras', Linear Alg. App. 183 (1993) 223-236.4. S. GONZALEZ and C. MARTINEZ, 'Idempotent elements in a Bernstein algebra', J. London Math. Soc.

(2) 42 (1990) 43(M36.5. I. R. HENTZEL and L. A. PERESI, 'Semiprime Bernstein algebras', Arch. Math. 52 (1989) 539-543.6. P. HOLGATE, 'Genetic algebras satisfying Bernstein's stationary principle', J. London Math. Soc. (2)

9(1975)613-623.7. N. JACOBSON, Lectures in abstract algebra I (Springer, New York-Heidelberg-Berlin, 1951).8. Yu. I. LYUBICH, 'Basic concepts and theorems of the evolutionary genetics of free populations',

Russian Math. Surveys 26 (1971) 51-123.9. Yu. I. LYUBICH, 'Classification of some types of Bernstein algebras', Vestnik Kharkov. Gos. Univ.

205, vyp. 45 (1980) 124-137; Selecta Math. Soviet 6 (1987) 1-14.10. M. OUATTARA, 'Algebres de Jordan et algebres genetiques', Cahiers Math. 37, (1988).11. S. WALCHER, 'Bernstein algebras which are Jordan algebras', Arch. Math. 50 (1988) 218-222.12. A. WORZ-BUSEKROS, Algebras in genetics, Lecture Notes in Biomathematics 36 (Springer, Berlin-

Heidelberg, 1980).13. A. WORZ-BUSEKROS, 'Bernstein algebras', Arch. Math. 48 (1987) 388-398.

Departamento de MatematicasFacultad de CienciasUniversidad de OviedoCalvo Sotelo s/n33007 OviedoSpain

Departamento de MatematicasFacultad de CienciasUniversidad de Zaragoza50009 ZaragozaSpain

on the structure of bernstein algebrasbrusso/cortesmontanerjlms95bis.pdfsections 2 and 3 suggest the...

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